I’ve been reading a lot about massive stellar objects, degenerate matter, and how the Pauli exclusion principle works at that scale. One thing I don’t understand is what it means for two particles to occupy the same quantum state, or what a quantum state really is.

My background in computers probably isn’t helping either. When I think of what “state” means, I imagine a class or a structure. It has a spin field, an energy_level field, and whatever else is required by the model. Two such instances would be indistinguishable if all of their properties were equal. Is this in any way relevant to what a quantum state is, or should I completely abandon this idea?

How many properties does it take to describe, for example, an electron? What kind of precision does it take to tell whether the two states are identical?

Is it even possible to explain it in an intuitive manner?

  • CanadaPlus@lemmy.sdf.orgEnglish
    1·
    7 days ago

    Is this calculated by assuming the wavefunction is static? Like, maybe a steady-state eigenfunction of the system’s evolution with an eigenvalue that’s 1, or another root of unity.

    • FrenziedFelidFanatic@pawb.socialEnglish
      1·
      7 days ago

      Is this calculated by assuming the wavefunction is static?

      Typically sorta? The way the Schrödinger equation is typically solved is by taking linear combinations of eigenfunctions (of the Hamiltonian) and making them time-dependent with a time-dependent phase out front.

      The eigenfunctions are otherwise time-independent since you can usually make the Hamiltonian be time independent.

      If the problem is easier to think about with a time-dependent Hamiltonian, you can use the Heisenberg formulation of quantum mechanics, which makes the wavefunctions static and lets the operators evolve in time. This can be helpful in a number of situations—typically involving light.

      Like, maybe a steady-state eigenfunction of the system’s evolution with an eigenvalue that’s 1, or another root of unity.

      I assume you mean eigenfunction of the Hamiltonian here, but the eigenvalue associated with that eigenfunction would be the energy of the state, so you can’t really make it be a root of unity (it must, in fact, be fully real since energy is an observable)

      • CanadaPlus@futurology.todayEnglish
        1·
        2 days ago

        I’ll admit, I only have a fuzzy understanding of even the basics of Hamiltonian mechanics. I understand quantum computing, though, and that evolution of a circuit is a unitary (linear) operator/matrix. So, wouldn’t continuous evolution be a one-parameter Lie subgroup of the unitary operators over your Hilbert space? Any eigenvalue would have to be a root of unity, with the exact one corresponding to rate of change in phase, because otherwise you end up with probabilities not summing to 1.

        I think it would be analogous to the normal modes for a classical standing wave, which are also used as examples of an eigenfunction.

        Maybe the more relevant question is if nonequilibrium, dynamical quantum systems can also be said to be quantised in the same way. Can they?

        If the problem is easier to think about with a time-dependent Hamiltonian, you can use the Heisenberg formulation of quantum mechanics, which makes the wavefunctions static and lets the operators evolve in time. This can be helpful in a number of situations—typically involving light.

        That sounds wild!